Modeling still matters: a surprising instance of catastrophic floating point errors in mathematical biology and numerical methods for ODEs

We guide the reader on a journey through mathematical modeling and numerical analysis, emphasizing the crucial interplay of both disciplines. Targeting undergraduate students with basic knowledge in dynamical systems and numerical methods for ordinary differential equations, we explore a model from mathematical biology where numerical methods fail badly due to catastrophic floating point errors. We analyze the reasons for this behavior by studying the steady states of the model and use the theory of invariants to develop an alternative model that is suited for numerical simulations. Our story intends to motivate combining analytical and numerical knowledge, even in cases where the world looks fine at first sight. We have set up an online repository containing an interactive notebook with all numerical experiments to make this study fully reproducible and useful for classroom teaching.Comment: 17 pages, 10 figure

Enhancing accuracy for solving American CEV model with high-order compact scheme and adaptive time stepping

In this research work, we propose a high-order time adapted scheme for pricing a coupled system of fixed-free boundary constant elasticity of variance (CEV) model on both equidistant and locally refined space-grid. The performance of our method is substantially enhanced to improve irregularities in the model which are both inherent and induced. Furthermore, the system of coupled PDEs is strongly nonlinear and involves several time-dependent coefficients that include the first-order derivative of the early exercise boundary. These coefficients are approximated from a fourth-order analytical approximation which is derived using a regularized square-root function. The semi-discrete equation for the option value and delta sensitivity is obtained from a non-uniform fourth-order compact finite difference scheme. Fifth-order 5(4) Dormand-Prince time integration method is used to solve the coupled system of discrete equations. Enhancing the performance of our proposed method with local mesh refinement and adaptive strategies enables us to obtain highly accurate solution with very coarse space grids, hence reducing computational runtime substantially. We further verify the performance of our methodology as compared with some of the well-known and better-performing existing methods

Numerical solutions for lunar orbits

Starting from a variational formulation based on Hamilton’s Principle, the paper exploits the finite element technique in the time domain in order to solve orbital dynamic problems characterised by constrained boundary value rather than initial value problems. The solution is obtained assembling a suitable number of finite elements inside the time interval of interest, imposing the desired constraints, and solving the resultant set of non-linear algebraic equations by means of Newton-Raphson method. In particular, in this work this general solution strategy is applied to periodic orbits determination. The effectiveness of the approach in finding periodic orbits in the unhomogeneous gravity field of the Moon is assessed by means of relevant examples, and the results are compared with those obtained by standard time marching techniques as well as with analytical results

Better Integrators for Functional Renormalization Group Calculations

We analyze a variety of integration schemes for the momentum space functional renormalization group calculation with the goal of finding an optimized scheme. Using the square lattice $t-t'$ Hubbard model as a testbed we define and benchmark the quality. Most notably we define an error estimate of the solution for the ordinary differential equation circumventing the issues introduced by the divergences at the end of the FRG flow. Using this measure to control for accuracy we find a threefold reduction in number of required integration steps achievable by choice of integrator. We herewith publish a set of recommended choices for the functional renormalization group, shown to decrease the computational cost for FRG calculations and representing a valuable basis for further investigations.Comment: 13 pages, 5 figure

Translating parameter estimation problems from EASY-FIT to SOCS

Mathematical models often involve unknown parameters that must be fit to experimental data. These so-called parameter estimation problems have many applications that may involve differential equations, optimization, and control theory. EASY-FIT and SOCS are two software packages that solve parameter estimation problems. In this thesis, we discuss the design and implementation of a source-to-source translator called EFtoSOCS used to translate EASY FIT input into SOCS input. This makes it possible to test SOCS on a large number of parameter estimation problems available in the EASY-FIT problem database that vary both in size and difficulty.Parameter estimation problems typically have many locally optimal solutions, and the solution obtained often depends critically on the initial guess for the solution. A 3-stage approach is followed to enhance the convergence of solutions in SOCS. The stages are designed to use an initial guess that is progressively closer to the optimal solution found by EASY-FIT. Using this approach we run EFtoSOCS on all translatable problems (691) from the EASY-FIT database. We find that all but 7 problems produce converged solutions in SOCS. We describe the reasons that SOCS was not able solve these problems, compare the solutions found by SOCS and EASY-FIT, and suggest possible improvements to both EFtoSOCS and SOCS